L(s) = 1 | + 1.96·2-s + 0.187·3-s + 1.84·4-s + 0.368·6-s − 0.677·7-s − 0.297·8-s − 2.96·9-s + 2.84·11-s + 0.346·12-s + 4.76·13-s − 1.32·14-s − 4.27·16-s − 5.18·17-s − 5.81·18-s − 0.127·21-s + 5.57·22-s − 1.05·23-s − 0.0557·24-s + 9.35·26-s − 1.11·27-s − 1.25·28-s + 1.42·29-s − 0.271·31-s − 7.80·32-s + 0.533·33-s − 10.1·34-s − 5.48·36-s + ⋯ |
L(s) = 1 | + 1.38·2-s + 0.108·3-s + 0.924·4-s + 0.150·6-s − 0.255·7-s − 0.105·8-s − 0.988·9-s + 0.856·11-s + 0.100·12-s + 1.32·13-s − 0.354·14-s − 1.06·16-s − 1.25·17-s − 1.37·18-s − 0.0277·21-s + 1.18·22-s − 0.219·23-s − 0.0113·24-s + 1.83·26-s − 0.215·27-s − 0.236·28-s + 0.264·29-s − 0.0486·31-s − 1.37·32-s + 0.0928·33-s − 1.74·34-s − 0.913·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 3 | \( 1 - 0.187T + 3T^{2} \) |
| 7 | \( 1 + 0.677T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 - 1.42T + 29T^{2} \) |
| 31 | \( 1 + 0.271T + 31T^{2} \) |
| 37 | \( 1 - 0.603T + 37T^{2} \) |
| 41 | \( 1 + 6.73T + 41T^{2} \) |
| 43 | \( 1 - 5.62T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 - 6.88T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 - 7.60T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.90643831539690555154512848105, −6.56483276852585626128172468732, −5.89852222389734984886913332933, −5.40111638058548573547856595062, −4.41129906256334726695501635628, −3.95346807247367999692441465338, −3.22771373941875389413995155848, −2.57559184851321350187082898176, −1.51478931190251268560397175956, 0,
1.51478931190251268560397175956, 2.57559184851321350187082898176, 3.22771373941875389413995155848, 3.95346807247367999692441465338, 4.41129906256334726695501635628, 5.40111638058548573547856595062, 5.89852222389734984886913332933, 6.56483276852585626128172468732, 6.90643831539690555154512848105